Problem

In a one-dimensional flow field, the velocity at a point may be given in the Eulerian system by u=x+2t. Determine the displacement of a fluid particle whose initial position is x₀ at initial time to in the Lagrangian system.

Solution

  Given u=x+2t

gives on integration

	(A)

where, c is the constant of integration. C can be found by initial conditions-

For

Therefore,  

	(B)

Putting value of c from (B) into (A)

This equation represents the Lagrangian version of the fluid particle having the identity at